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Supply the two endpoint factors
The fictional endpoints are D1 = 0.96 and D2 = 0.90. A payment of 100 is due at 1.5 years. No midpoint factor is supplied, so the chosen rule must be stated rather than treated as observed data.
Interpolate logarithms of factors
Halfway between the log factors, ln(D1.5) = [ln(0.96)+ln(0.90)]/2. Therefore D1.5 = √(0.96×0.90) = 0.929516. Its payment value is 92.9516.
Interpolate annual effective zero rates
The endpoint annual zeros are 0.96^−1−1 and 0.90^−1/2−1. Their arithmetic midpoint is 4.787961%. Applying that rate for 1.5 years gives factor 0.932251 and payment value 93.225112.
| Rule | Midpoint factor | Value of 100 |
|---|---|---|
| Linear log factor | 0.929516 | 92.9516 |
| Linear annual effective zero | 0.932251 | 93.225112 |
Report the model difference
Neither result comes from a quoted midpoint instrument in this case. The difference is caused by the interpolation target. Preserve enough precision to observe it; rounding both outputs early can hide the consequence. A suitable market method may need additional fitting and smoothness constraints.